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Matrices arising in a discrete-time deterministic model of population growth [[#References|[a3]]]. The Leslie model considers individuals of one sex in a population which is closed to migration. The maximum life span is $k$ time units, and an individual is said to be in the $i$th age group if its exact age falls in the interval $[ i - 1 , i )$, for some $1 \leq i \leq k$. The corresponding Leslie matrix is given by

\begin{equation*} L = \left( \begin{array} { c c c c c } { m _ { 1 } } &amp; { m _ { 2 } } &amp; { \ldots } &amp; { \ldots } &amp; { m _ { k } } \\ { p _ { 1 } } &amp; { 0 } &amp; { \ldots } &amp; { \ldots } &amp; { 0 } \\ { 0 } &amp; { p _ { 2 } } &amp; { 0 } &amp; { \ldots } &amp; { 0 } \\ { \vdots } &amp; { } &amp; { \ddots } &amp; {  } &amp; { \vdots } \\ { 0 } &amp; { \ldots } &amp; { 0 } &amp; { p _ { k - 1 } } &amp; { 0 } \end{array} \right), \end{equation*}

where for each $1 \leq i \leq k - 1$, $p _ { i }$ is the proportion of individuals in the $i$th age group who survive one time unit (this is assumed to be positive), and for each $1 \leq i \leq k$, $m_i$ is the average number of individuals produced in one time unit by a member of the $i$th age group. Let $v _ { i , t }$ be the average number of individuals in the $i$th age group at time $t$ units, and let $v _ { t }$ be the vector

\begin{equation*} \left( \begin{array} { c } { v _ { 1 , t }} \\ { \vdots } \\ { v _ { k , t } } \end{array} \right). \end{equation*}

Then $v _ { t  + 1} = L v_ t $, and since the conditions of mortality and fertility are assumed to persist, $v _ { t } = L ^ { t } v _ { 0 }$ for each integer $t \geq 0$.

If some $m_i$ is positive, then $L$ has one positive [[Eigen value|eigen value]] $r$ which is a simple root of the [[Characteristic polynomial|characteristic polynomial]]. For any eigenvalue $\lambda$ of $L$, $r \geq | \lambda |$; indeed $L$ has exactly $d$ eigenvalues of modulus $r$, where $d$ is the [[Greatest common divisor|greatest common divisor]] of $\{ i : m_i > 0 \}$. Corresponding to the eigenvalue $r$ is the right eigenvector $w$ given by the formula

\begin{equation*} w = \frac { 1 } { s } \left( \begin{array} { c } { 1 } \\ { p _ { 1 } / r } \\ { p _ { 1 } p _ { 2 } / r ^ { 2 } } \\ { \vdots } \\ { p _ { 1 } \dots p _ { k - 1}  / r ^ { k - 1 } } \end{array} \right), \end{equation*}

where $s = 1 + p _ { 1 } / r + \ldots + p _ { 1 } \ldots p _ { k - 1 } / r ^ { k - 1 }$. A left eigenvector corresponding to $r$ has the form $[ y _ { 1 } \ldots y _ { k } ]$, where for $1 \leq j \leq k$,

\begin{equation*} y _ { j } = \sum _ { i = j } ^ { k } p _ { j } \ldots p _ { i - 1 } m _ { i } r ^ { j - i - 1 }. \end{equation*}

The quantity $y_j$ is interpreted as the reproductive value of an individual in the $j$th age group.

Suppose that there are indices $i$, $j$ such that $1 \leq i \leq j \leq k$, and both $m_j$ and $v _ { i ,0} $ are positive. If $d > 1$, the sequence of age-distribution vectors, $v _ { t } / \sum _ { i = 1 } ^ { k } v _ { i , t }$, is asymptotically periodic as $t \rightarrow \infty$, and the period is a divisor of $d$ depending on $v_0$. When $d = 1$, then as $t \rightarrow \infty$, the sequence of age-distribution vectors converges to the eigenvector $w$, which is called the asymptotic stable age distribution for the population. The nature of the convergence of the age distributions is governed by the quantities $\lambda / r$, where $\lambda$ is an eigenvalue of $L$ distinct from $r$; a containment region in the complex plane for these quantities has been characterized (cf. [[#References|[a2]]], [[#References|[a5]]]). The sequence of vectors $v _ { t }$ is asymptotic to $c r ^ { t } w$, where $c$ is a positive constant depending on $v_0$; hence $r$ is sometimes called the rate of increase for the population. The sensitivity of $r$ to changes in $L$ is discussed in [[#References|[a1]]] and [[#References|[a6]]].

Variations on the Leslie model include matrix models for populations classified by criteria other than age (see [[#References|[a1]]]), and a model involving a sequence of Leslie matrices changing over time (see [[#References|[a4]]] and [[#References|[a6]]]). A stochastic version of the Leslie model yields a convergence result for the sequence $v _ { t } / r ^ { t }$ under the hypotheses that $d = 1$ and $r > 1$ (see [[#References|[a6]]]).

====References====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  H. Caswell,   "Matrix population models" , Sinauer  (1989)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  K.P. Hadeler,   G. Meinardus,   "On the roots of Cauchy polynomials"  ''Linear Alg. &amp; Its Appl.'' , '''38'''  (1981)  pp. 81–102</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P.H. Leslie,   "On the use of matrices in certain population mathematics"  ''Biometrika'' , '''33'''  (1945)  pp. 213–245</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  N. Keyfitz,   "Introduction to the mathematics of population" , Addison-Wesley  (1977)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  S. Kirkland,   "An eigenvalue region for Leslie matrices"  ''SIAM J. Matrix Anal. Appl.'' , '''13'''  (1992)  pp. 507–529</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  J.H. Pollard,   "Mathematical models for the growth of human populations" , Cambridge Univ. Press  (1973)</td></tr>
</table>